How can you establish a relationship between a distant music image and the epiphany in "The Dead"?

What precisely are you referring to by the "distant music image"? I wonder if you are talking about the last line of the text:

His soul swooned slowly as he heard the snow falling faintly through the universe and faintly falling, like the descent of their last end, upon all the living and the dead.

It doesn't actually seem to be a "music image" as you suggest, but it does talk about the sound of the snow...

What precisely are you referring to by the "distant music image"? I wonder if you are talking about the last line of the text:

His soul swooned slowly as he heard the snow falling faintly through the universe and faintly falling, like the descent of their last end, upon all the living and the dead.

It doesn't actually seem to be a "music image" as you suggest, but it does talk about the sound of the snow falling and Gabriel Conroy hearing that sound. If we are thinking about how this last image of the sound of the snow falling and covering everyone, both the "living and the dead," then we need to be aware of how this is linked in with the epiphany that Gabriel Conroy experiences in the last page or so of this tremendous story. Having heard his wife's confession of her romance with Michael Furey, he is forced to confront the fact that he has never really loved:

He had never felt like that himself towards any woman but he knew that such a feeling must be love.

In addition, he muses on the way that all of us are going to die and how we should live our lives as a result:

Better pass boldly into that other world, in the full glory of some passion, than fade and wither dismally with age.

The existence of Michael Furey, who lived a life of passion (consider his last name), stands as a rebuke to Gabriel Conroy, who doesn't really "feel" and seems ridden with paralysis throughout the story that prevents him from acting in so many ways. Having experienced this insight into his own life and how life should be lived, we return to the image of the snow, falling equally over "all the living and the dead," reminding us once more of our eventual destiny but also challenging us to live as if we were "alive" rather than someone, like Gabriel Conroy, who is already living as if they were "dead."